Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.8% → 99.6%

Time: 4.1s

Alternatives: 16

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ (- z y) (- t (- z 1.0))) a x))

C

double code(double x, double y, double z, double t, double a) {
	return fma(((z - y) / (t - (z - 1.0))), a, x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(Float64(z - y) / Float64(t - Float64(z - 1.0))), a, x)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(z - y), $MachinePrecision] / N[(t - N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

   4. sub-div N/A

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

   6. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

   7. associate--l+ N/A

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

   9. associate-+l- N/A

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   11. lower--.f64 99.6

       \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - z}{\frac{t}{a}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7200000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (- y z) (/ t a)))))
   (if (<= t -2e+149)
     t_1
     (if (<= t 7200000000000.0) (fma (/ (- z y) (- 1.0 z)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) / (t / a));
	double tmp;
	if (t <= -2e+149) {
		tmp = t_1;
	} else if (t <= 7200000000000.0) {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y - z) / Float64(t / a)))
	tmp = 0.0
	if (t <= -2e+149)
		tmp = t_1;
	elseif (t <= 7200000000000.0)
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y - z), $MachinePrecision] / N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+149], t$95$1, If[LessEqual[t, 7200000000000.0], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.0000000000000001e149 or 7.2e12 < t

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in t around inf

      \[\leadsto x - \frac{y - z}{\color{blue}{\frac{t}{a}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 53.5

         \[\leadsto x - \frac{y - z}{\frac{t}{\color{blue}{a}}} \]

   4. Applied rewrites 53.5%

      \[\leadsto x - \frac{y - z}{\color{blue}{\frac{t}{a}}} \]

   if -2.0000000000000001e149 < t < 7.2e12

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{-z}, a, x\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1700000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{1 + t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- z)) a x)))
   (if (<= z -1e+35)
     t_1
     (if (<= z 1700000.0) (fma (/ (- y) (+ 1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / -z), a, x);
	double tmp;
	if (z <= -1e+35) {
		tmp = t_1;
	} else if (z <= 1700000.0) {
		tmp = fma((-y / (1.0 + t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(-z)), a, x)
	tmp = 0.0
	if (z <= -1e+35)
		tmp = t_1;
	elseif (z <= 1700000.0)
		tmp = fma(Float64(Float64(-y) / Float64(1.0 + t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / (-z)), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -1e+35], t$95$1, If[LessEqual[z, 1700000.0], N[(N[((-y) / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999997e34 or 1.7e6 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{-1 \cdot z}, a, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(z\right)}, a, x\right) \]

      2. lower-neg.f64 59.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{-z}, a, x\right) \]

   7. Applied rewrites 59.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{-z}, a, x\right) \]

   if -9.9999999999999997e34 < z < 1.7e6

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{y}{1 + t}, a, x\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{1 + t}, a, x\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{1 + t}, a, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(y\right)}{1 + t}, a, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-y}{1 + t}, a, x\right) \]

      5. lift-+.f64 71.6

         \[\leadsto \mathsf{fma}\left(\frac{-y}{1 + t}, a, x\right) \]

   7. Applied rewrites 71.6%

      \[\leadsto \mathsf{fma}\left(\frac{-y}{1 + t}, a, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{-z}, a, x\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1700000:\\ \;\;\;\;x - a \cdot \frac{y}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- z)) a x)))
   (if (<= z -1e+35)
     t_1
     (if (<= z 1700000.0) (- x (* a (/ y (+ 1.0 t)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / -z), a, x);
	double tmp;
	if (z <= -1e+35) {
		tmp = t_1;
	} else if (z <= 1700000.0) {
		tmp = x - (a * (y / (1.0 + t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(-z)), a, x)
	tmp = 0.0
	if (z <= -1e+35)
		tmp = t_1;
	elseif (z <= 1700000.0)
		tmp = Float64(x - Float64(a * Float64(y / Float64(1.0 + t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / (-z)), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -1e+35], t$95$1, If[LessEqual[z, 1700000.0], N[(x - N[(a * N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999997e34 or 1.7e6 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{-1 \cdot z}, a, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(z\right)}, a, x\right) \]

      2. lower-neg.f64 59.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{-z}, a, x\right) \]

   7. Applied rewrites 59.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{-z}, a, x\right) \]

   if -9.9999999999999997e34 < z < 1.7e6

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 71.6

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 71.6%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+39}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1700000:\\ \;\;\;\;x - a \cdot \frac{y}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+39)
   (- x a)
   (if (<= z 1700000.0) (- x (* a (/ y (+ 1.0 t)))) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+39) {
		tmp = x - a;
	} else if (z <= 1700000.0) {
		tmp = x - (a * (y / (1.0 + t)));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1d+39)) then
        tmp = x - a
    else if (z <= 1700000.0d0) then
        tmp = x - (a * (y / (1.0d0 + t)))
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+39) {
		tmp = x - a;
	} else if (z <= 1700000.0) {
		tmp = x - (a * (y / (1.0 + t)));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e+39:
		tmp = x - a
	elif z <= 1700000.0:
		tmp = x - (a * (y / (1.0 + t)))
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+39)
		tmp = Float64(x - a);
	elseif (z <= 1700000.0)
		tmp = Float64(x - Float64(a * Float64(y / Float64(1.0 + t))));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e+39)
		tmp = x - a;
	elseif (z <= 1700000.0)
		tmp = x - (a * (y / (1.0 + t)));
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+39], N[(x - a), $MachinePrecision], If[LessEqual[z, 1700000.0], N[(x - N[(a * N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999994e38 or 1.7e6 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if -9.9999999999999994e38 < z < 1.7e6

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 71.6

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 71.6%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-221}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{elif}\;t \leq 0.000116:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -8e+18)
     t_1
     (if (<= t -5.5e-221)
       (fma (/ z (- 1.0 z)) a x)
       (if (<= t 0.000116) (fma (/ (- z y) 1.0) a x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -8e+18) {
		tmp = t_1;
	} else if (t <= -5.5e-221) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else if (t <= 0.000116) {
		tmp = fma(((z - y) / 1.0), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -8e+18)
		tmp = t_1;
	elseif (t <= -5.5e-221)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	elseif (t <= 0.000116)
		tmp = fma(Float64(Float64(z - y) / 1.0), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -8e+18], t$95$1, If[LessEqual[t, -5.5e-221], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[t, 0.000116], N[(N[(N[(z - y), $MachinePrecision] / 1.0), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8e18 or 1.16e-4 < t

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

      2. lift--.f64 52.9

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

   7. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

   if -8e18 < t < -5.49999999999999966e-221

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.3%

       \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]

   if -5.49999999999999966e-221 < t < 1.16e-4

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1}, a, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.3%

       \[\leadsto \mathsf{fma}\left(\frac{z - y}{1}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-221}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -8e+18)
     t_1
     (if (<= t -5.5e-221)
       (fma (/ z (- 1.0 z)) a x)
       (if (<= t 9.5e-21) (fma (/ (- y) 1.0) a x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -8e+18) {
		tmp = t_1;
	} else if (t <= -5.5e-221) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else if (t <= 9.5e-21) {
		tmp = fma((-y / 1.0), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -8e+18)
		tmp = t_1;
	elseif (t <= -5.5e-221)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	elseif (t <= 9.5e-21)
		tmp = fma(Float64(Float64(-y) / 1.0), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -8e+18], t$95$1, If[LessEqual[t, -5.5e-221], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[t, 9.5e-21], N[(N[((-y) / 1.0), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8e18 or 9.4999999999999994e-21 < t

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

      2. lift--.f64 52.9

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

   7. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

   if -8e18 < t < -5.49999999999999966e-221

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.3%

       \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]

   if -5.49999999999999966e-221 < t < 9.4999999999999994e-21

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{1 - z}, a, x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(y\right)}{1 - z}, a, x\right) \]

      2. lower-neg.f64 64.2

         \[\leadsto \mathsf{fma}\left(\frac{-y}{1 - z}, a, x\right) \]

   10. Applied rewrites 64.2%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1 - z}, a, x\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1}, a, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 56.5%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-102}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -2.9e-5)
     t_1
     (if (<= t -3.1e-102)
       (- x a)
       (if (<= t 9.5e-21) (fma (/ (- y) 1.0) a x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -2.9e-5) {
		tmp = t_1;
	} else if (t <= -3.1e-102) {
		tmp = x - a;
	} else if (t <= 9.5e-21) {
		tmp = fma((-y / 1.0), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -2.9e-5)
		tmp = t_1;
	elseif (t <= -3.1e-102)
		tmp = Float64(x - a);
	elseif (t <= 9.5e-21)
		tmp = fma(Float64(Float64(-y) / 1.0), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -2.9e-5], t$95$1, If[LessEqual[t, -3.1e-102], N[(x - a), $MachinePrecision], If[LessEqual[t, 9.5e-21], N[(N[((-y) / 1.0), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.9e-5 or 9.4999999999999994e-21 < t

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

      2. lift--.f64 52.9

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

   7. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{t}, a, x\right) \]

   if -2.9e-5 < t < -3.10000000000000013e-102

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if -3.10000000000000013e-102 < t < 9.4999999999999994e-21

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{1 - z}, a, x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(y\right)}{1 - z}, a, x\right) \]

      2. lower-neg.f64 64.2

         \[\leadsto \mathsf{fma}\left(\frac{-y}{1 - z}, a, x\right) \]

   10. Applied rewrites 64.2%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1 - z}, a, x\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1}, a, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 56.5%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+34}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 550000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.2e+34)
   (- x a)
   (if (<= z 550000.0) (fma (/ (- y) 1.0) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e+34) {
		tmp = x - a;
	} else if (z <= 550000.0) {
		tmp = fma((-y / 1.0), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.2e+34)
		tmp = Float64(x - a);
	elseif (z <= 550000.0)
		tmp = fma(Float64(Float64(-y) / 1.0), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.2e+34], N[(x - a), $MachinePrecision], If[LessEqual[z, 550000.0], N[(N[((-y) / 1.0), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999993e34 or 5.5e5 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if -9.1999999999999993e34 < z < 5.5e5

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot a + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}, \color{blue}{a}, x\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1 + t\right) - z}, a, x\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 + \left(t - z\right)}, a, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

      11. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{t - \left(z - 1\right)}, a, x\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]
   6. Step-by-step derivation

      1. lower--.f64 80.3

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1 \cdot y}{1 - z}, a, x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(y\right)}{1 - z}, a, x\right) \]

      2. lower-neg.f64 64.2

         \[\leadsto \mathsf{fma}\left(\frac{-y}{1 - z}, a, x\right) \]

   10. Applied rewrites 64.2%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1 - z}, a, x\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1}, a, x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 56.5%

       \[\leadsto \mathsf{fma}\left(\frac{-y}{1}, a, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+34}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 550000:\\ \;\;\;\;x - a \cdot \frac{y}{1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.2e+34)
   (- x a)
   (if (<= z 550000.0) (- x (* a (/ y 1.0))) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e+34) {
		tmp = x - a;
	} else if (z <= 550000.0) {
		tmp = x - (a * (y / 1.0));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-9.2d+34)) then
        tmp = x - a
    else if (z <= 550000.0d0) then
        tmp = x - (a * (y / 1.0d0))
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e+34) {
		tmp = x - a;
	} else if (z <= 550000.0) {
		tmp = x - (a * (y / 1.0));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.2e+34:
		tmp = x - a
	elif z <= 550000.0:
		tmp = x - (a * (y / 1.0))
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.2e+34)
		tmp = Float64(x - a);
	elseif (z <= 550000.0)
		tmp = Float64(x - Float64(a * Float64(y / 1.0)));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.2e+34)
		tmp = x - a;
	elseif (z <= 550000.0)
		tmp = x - (a * (y / 1.0));
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.2e+34], N[(x - a), $MachinePrecision], If[LessEqual[z, 550000.0], N[(x - N[(a * N[(y / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999993e34 or 5.5e5 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if -9.1999999999999993e34 < z < 5.5e5

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 71.6

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 71.6%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]

   5. Taylor expanded in t around 0

      \[\leadsto x - a \cdot \frac{y}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.5%

      \[\leadsto x - a \cdot \frac{y}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+35}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3050000:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+35)
   (- x a)
   (if (<= z 3050000.0) (- x (* a (/ y t))) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+35) {
		tmp = x - a;
	} else if (z <= 3050000.0) {
		tmp = x - (a * (y / t));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.02d+35)) then
        tmp = x - a
    else if (z <= 3050000.0d0) then
        tmp = x - (a * (y / t))
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+35) {
		tmp = x - a;
	} else if (z <= 3050000.0) {
		tmp = x - (a * (y / t));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e+35:
		tmp = x - a
	elif z <= 3050000.0:
		tmp = x - (a * (y / t))
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+35)
		tmp = Float64(x - a);
	elseif (z <= 3050000.0)
		tmp = Float64(x - Float64(a * Float64(y / t)));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.02e+35)
		tmp = x - a;
	elseif (z <= 3050000.0)
		tmp = x - (a * (y / t));
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+35], N[(x - a), $MachinePrecision], If[LessEqual[z, 3050000.0], N[(x - N[(a * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000007e35 or 3.05e6 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if -1.02000000000000007e35 < z < 3.05e6

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 71.6

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 71.6%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]

   5. Taylor expanded in t around inf

      \[\leadsto x - a \cdot \frac{y}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 55.2

         \[\leadsto x - a \cdot \frac{y}{t} \]

   7. Applied rewrites 55.2%

      \[\leadsto x - a \cdot \frac{y}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+35}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3050000:\\ \;\;\;\;x - \frac{a \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+35)
   (- x a)
   (if (<= z 3050000.0) (- x (/ (* a y) t)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+35) {
		tmp = x - a;
	} else if (z <= 3050000.0) {
		tmp = x - ((a * y) / t);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.02d+35)) then
        tmp = x - a
    else if (z <= 3050000.0d0) then
        tmp = x - ((a * y) / t)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+35) {
		tmp = x - a;
	} else if (z <= 3050000.0) {
		tmp = x - ((a * y) / t);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e+35:
		tmp = x - a
	elif z <= 3050000.0:
		tmp = x - ((a * y) / t)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+35)
		tmp = Float64(x - a);
	elseif (z <= 3050000.0)
		tmp = Float64(x - Float64(Float64(a * y) / t));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.02e+35)
		tmp = x - a;
	elseif (z <= 3050000.0)
		tmp = x - ((a * y) / t);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+35], N[(x - a), $MachinePrecision], If[LessEqual[z, 3050000.0], N[(x - N[(N[(a * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000007e35 or 3.05e6 < z

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if -1.02000000000000007e35 < z < 3.05e6

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 71.6

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 71.6%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]

   5. Taylor expanded in t around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lift-*.f64 53.3

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 53.3%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+267}:\\ \;\;\;\;a \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_1 -1e+267)
     (* a (/ y z))
     (if (<= t_1 INFINITY) (- x a) (/ (* a y) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -1e+267) {
		tmp = a * (y / z);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x - a;
	} else {
		tmp = (a * y) / z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -1e+267) {
		tmp = a * (y / z);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x - a;
	} else {
		tmp = (a * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_1 <= -1e+267:
		tmp = a * (y / z)
	elif t_1 <= math.inf:
		tmp = x - a
	else:
		tmp = (a * y) / z
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_1 <= -1e+267)
		tmp = Float64(a * Float64(y / z));
	elseif (t_1 <= Inf)
		tmp = Float64(x - a);
	else
		tmp = Float64(Float64(a * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_1 <= -1e+267)
		tmp = a * (y / z);
	elseif (t_1 <= Inf)
		tmp = x - a;
	else
		tmp = (a * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+267], N[(a * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x - a), $MachinePrecision], N[(N[(a * y), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -9.9999999999999997e266

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot y}{\left(1 + t\right) - z}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot y\right)}{\color{blue}{\left(1 + t\right) - z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot y\right)}{\color{blue}{\left(1 + t\right) - z}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a \cdot y\right)}{\color{blue}{\left(1 + t\right)} - z} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-a \cdot y}{\color{blue}{\left(1 + t\right)} - z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-a \cdot y}{\left(\color{blue}{1} + t\right) - z} \]

      6. associate--l+ N/A

         \[\leadsto \frac{-a \cdot y}{1 + \color{blue}{\left(t - z\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{-a \cdot y}{\left(t - z\right) + \color{blue}{1}} \]

      8. associate-+l- N/A

         \[\leadsto \frac{-a \cdot y}{t - \color{blue}{\left(z - 1\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{-a \cdot y}{t - \color{blue}{\left(z - 1\right)}} \]

      10. lower--.f64 24.0

          \[\leadsto \frac{-a \cdot y}{t - \left(z - \color{blue}{1}\right)} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{-a \cdot y}{t - \left(z - 1\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      2. lift-*.f64 8.7

         \[\leadsto \frac{a \cdot y}{z} \]

   7. Applied rewrites 8.7%

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      3. associate-/l* N/A

         \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

      4. lower-*.f64 N/A

         \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

      5. lower-/.f64 9.8

         \[\leadsto a \cdot \frac{y}{z} \]

   9. Applied rewrites 9.8%

      \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

   if -9.9999999999999997e266 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < +inf.0

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]

   if +inf.0 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot y}{\left(1 + t\right) - z}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot y\right)}{\color{blue}{\left(1 + t\right) - z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot y\right)}{\color{blue}{\left(1 + t\right) - z}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a \cdot y\right)}{\color{blue}{\left(1 + t\right)} - z} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-a \cdot y}{\color{blue}{\left(1 + t\right)} - z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-a \cdot y}{\left(\color{blue}{1} + t\right) - z} \]

      6. associate--l+ N/A

         \[\leadsto \frac{-a \cdot y}{1 + \color{blue}{\left(t - z\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{-a \cdot y}{\left(t - z\right) + \color{blue}{1}} \]

      8. associate-+l- N/A

         \[\leadsto \frac{-a \cdot y}{t - \color{blue}{\left(z - 1\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{-a \cdot y}{t - \color{blue}{\left(z - 1\right)}} \]

      10. lower--.f64 24.0

          \[\leadsto \frac{-a \cdot y}{t - \left(z - \color{blue}{1}\right)} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{-a \cdot y}{t - \left(z - 1\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      2. lift-*.f64 8.7

         \[\leadsto \frac{a \cdot y}{z} \]

   7. Applied rewrites 8.7%

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \frac{y}{z}\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (/ y z))) (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_2 -1e+267) t_1 (if (<= t_2 INFINITY) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / z);
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -1e+267) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / z);
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -1e+267) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * (y / z)
	t_2 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_2 <= -1e+267:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * Float64(y / z))
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -1e+267)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * (y / z);
	t_2 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_2 <= -1e+267)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+267], t$95$1, If[LessEqual[t$95$2, Infinity], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -9.9999999999999997e266 or +inf.0 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot y}{\left(1 + t\right) - z}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot y\right)}{\color{blue}{\left(1 + t\right) - z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot y\right)}{\color{blue}{\left(1 + t\right) - z}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a \cdot y\right)}{\color{blue}{\left(1 + t\right)} - z} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-a \cdot y}{\color{blue}{\left(1 + t\right)} - z} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-a \cdot y}{\left(\color{blue}{1} + t\right) - z} \]

      6. associate--l+ N/A

         \[\leadsto \frac{-a \cdot y}{1 + \color{blue}{\left(t - z\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{-a \cdot y}{\left(t - z\right) + \color{blue}{1}} \]

      8. associate-+l- N/A

         \[\leadsto \frac{-a \cdot y}{t - \color{blue}{\left(z - 1\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{-a \cdot y}{t - \color{blue}{\left(z - 1\right)}} \]

      10. lower--.f64 24.0

          \[\leadsto \frac{-a \cdot y}{t - \left(z - \color{blue}{1}\right)} \]

   4. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\frac{-a \cdot y}{t - \left(z - 1\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      2. lift-*.f64 8.7

         \[\leadsto \frac{a \cdot y}{z} \]

   7. Applied rewrites 8.7%

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{a \cdot y}{z} \]

      3. associate-/l* N/A

         \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

      4. lower-*.f64 N/A

         \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

      5. lower-/.f64 9.8

         \[\leadsto a \cdot \frac{y}{z} \]

   9. Applied rewrites 9.8%

      \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

   if -9.9999999999999997e266 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < +inf.0

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 60.2%

      \[\leadsto x - \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ x - a \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x a))

C

double code(double x, double y, double z, double t, double a) {
	return x - a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - a
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - a;
}

Python

def code(x, y, z, t, a):
	return x - a

Julia

function code(x, y, z, t, a)
	return Float64(x - a)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - a;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

2. Taylor expanded in z around inf

   \[\leadsto x - \color{blue}{a} \]
3. Step-by-step derivation

4. Applied rewrites 60.2%

   \[\leadsto x - \color{blue}{a} \]
5. Add Preprocessing

Alternative 16: 17.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ -a \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- a))

C

double code(double x, double y, double z, double t, double a) {
	return -a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -a
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -a;
}

Python

def code(x, y, z, t, a):
	return -a

Julia

function code(x, y, z, t, a)
	return Float64(-a)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -a;
end

Wolfram

code[x_, y_, z_, t_, a_] := (-a)

Derivation

1. Initial program 96.8%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot \color{blue}{a} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{z}{\left(1 + t\right) - z} - \frac{y}{\left(1 + t\right) - z}\right) \cdot \color{blue}{a} \]

   3. sub-div N/A

      \[\leadsto \frac{z - y}{\left(1 + t\right) - z} \cdot a \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{z - y}{\left(1 + t\right) - z} \cdot a \]

   5. lower--.f64 N/A

      \[\leadsto \frac{z - y}{\left(1 + t\right) - z} \cdot a \]

   6. associate--l+ N/A

      \[\leadsto \frac{z - y}{1 + \left(t - z\right)} \cdot a \]

   7. +-commutative N/A

      \[\leadsto \frac{z - y}{\left(t - z\right) + 1} \cdot a \]

   8. associate-+l- N/A

      \[\leadsto \frac{z - y}{t - \left(z - 1\right)} \cdot a \]

   9. lower--.f64 N/A

      \[\leadsto \frac{z - y}{t - \left(z - 1\right)} \cdot a \]

   10. lower--.f64 48.3

       \[\leadsto \frac{z - y}{t - \left(z - 1\right)} \cdot a \]

4. Applied rewrites 48.3%

   \[\leadsto \color{blue}{\frac{z - y}{t - \left(z - 1\right)} \cdot a} \]

5. Taylor expanded in z around inf

   \[\leadsto -1 \cdot \color{blue}{a} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(a\right) \]

   2. lower-neg.f64 17.5

      \[\leadsto -a \]

7. Applied rewrites 17.5%

   \[\leadsto -a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))